A non-amenable groupoid whose maximal and reduced $ C ...

arXiv:1504.05615v2 [math.OA] 22 May 2015

A NON-AMENABLE GROUPOID WHOSE MAXIMAL AND REDUCED C ∗ -ALGEBRAS ARE THE SAME RUFUS WILLETT

Abstract. We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and ´ etale.

1. Introduction Say G is a locally compact group equipped with a Haar measure, and Cc (G) is the convolution ∗-algebra of continuous, compactly supported, complex-valued functions on G. In general Cc (G) has many interesting C ∗ -algebra completions, ∗ but the two most important are: the maximal completion Cmax (G), which is the completion taken over (the integrated forms of) all unitary representations of G; ∗ and the reduced completion Cred (G), which is the completion of Cc (G) for the left 2 regular representation on L (G). An important theorem of Hulanicki [5] says that ∗ ∗ Cmax (G) = Cred (G) if and only if G is amenable. Now, say G is a locally compact, Hausdorff groupoid. To avoid measure-theoretic complications in the statements below, we will assume that G is ´etale. Much as in the case of groups, there is a canonically associated convolution ∗-algebra Cc (G) of continuous compactly supported functions on G, and this ∗-algebra completes in a ∗ ∗ (G). There is (G) and Cred natural way to maximal and reduced C ∗ -algebras Cmax also a natural definition of topological amenability due to Renault [11, discussion below Definition II.3.6] that generalises the definition for groups. See [11] and [1] for comprehensive treatments of groupoids and their C ∗ -algebras, and [13, Section 2.3] and [3, Section 5.6] for self-contained introductions covering only the (much simpler) ´etale case. For groupoids, one has the following analogue of one direction of Hulanicki’s theorem, for which we refer to book of Anantharaman-Delaroche and Renault [1, Proposition 3.3.5 and Proposition 6.1.8]1. Theorem 1.1. Say G is a locally compact, Hausdorff, ´etale groupoid. If G is ∗ ∗ (G) = Cred (G). topologically amenable, then Cmax In this note, we show that the converse to this result is false, so Hulanicki’s theorem does not extend to groupoids. 1Anantharaman-Delaroche and Renault actually prove this result in the much more sophisticated case that G is not ´ etale, and thus need additional assumptions about Haar systems. See [3, Corollary 5.6.17] for a self-contained proof of Theorem 1.1 as stated here. 1

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Theorem 1.2. There exist locally compact, Hausdorff, second countable, ´etale groupoids with compact unit space that are not topologically amenable, but that ∗ ∗ satisfy Cmax (G) = Cred (G). Remark 1.3. Briefly looking outside the ´etale case, there are (at least)2 two notions of amenability that are used for locally compact groupoids: topological amenability [1, Definition 2.2.8] and measurewise amenability [1, Definition 3.3.1]. For a general locally compact, second countable, Hausdorff groupoid G with continuous Haar system, one has the following implications G is topologically amenable ⇒ G is measurewise amenable ∗ ∗ (G) = Cred (G) ; ⇒ Cmax

for the first implication, see [1, Proposition 3.3.5], and for the second see [1, Proposition 6.1.8]. If G is a group (more generally, a transitive groupoid [16, Lemma 8]), all three of these conditions are equivalent. If G is ´etale, then it is automatically equipped with a continuous Haar system, and topological and measurewise amenability are equivalent [1, Corollary 3.3.8]. Thus our examples show that measurewise amenability does not imply equality of the maximal and reduced C ∗ -algebras of a groupoid. They do not, however, seem to have any bearing on the general relationship between topological and measurewise amenability. The examples used in Theorem 1.2 are a slight adaptation of counterexamples to the Baum-Connes conjecture for groupoids constructed by Higson, Lafforgue, and Skandalis [4, Section 2]. The essential extra ingredient needed is property FD of Lubotzky and Shalom [8]. Our examples are of a particularly simple form, and are in fact a bundle of groups: see Section 2 below for details of all this. The existence of examples as in Theorem 1.2 is a fairly well-known question (see for example [1, Remark 6.1.9], [13, Section 4.2], or [3, page 162]) and the answer did not seem to be known to experts. We thus thought Theorem 1.2 was worth publicizing despite the similarity to existing constructions. We should remark, however, that our results seem to have no bearing on the existence of transformation groupoids, or of principal groupoids, with the properties in Theorem 1.2. In particular, we cannot say anything about whether or not equality of maximal and reduced crossed products for a group action implies (topological) amenability of the action. See Section 3 for some comments along these lines. This paper is structured as follows. Section 2 recalls the construction of Higson, Lafforgue and Skandalis, and proves Theorem 1.2. The proof proceeds by characterizing when the class of examples studied by Higson, Lafforgue and Skandalis are amenable, and when their maximal and reduced groupoid C ∗ -algebras are the same; we then use property FD of Lubotzky and Shalom to show that examples ‘between’ these two properties exist. Section 3 collects some comments and questions about exactness, transformation groupoids, and coarse groupoids. Acknowledgements. The author was partially supported by the US NSF. The author is grateful to the Erwin Schr¨odinger Institute in Vienna for its support ˇ during part of the work on this paper, to Erik Guentner and J´an Spakula for 2Another definition is that of Borel amenability from [14, Definition 2.1]. The condition ∗ ∗ (G)’ could also be thought of as a form of amenability, and is sometimes (see ‘Cmax (G) = Cred e.g. [16]) called metric amenability.

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interesting conversations on related issues, and to Claire Anantharaman-Delaroche and Dana Williams for very useful comments on an earlier version of this note. 2. Main result We first recall a construction of a class of groupoids from [4, Section 2]; some groupoids from this class will have the properties in Theorem 1.2. The starting point for this construction is the following data. Definition 2.1. Let Γ be a discrete group. An approximating sequence for Γ is a sequence (Kn ) of subgroups of Γ with the following properties. (i) Each Kn is a normal, finite index subgroup of Γ. (ii) The sequence is nested: Kn ⊇ Kn+1 for allT n. (iii) The intersection of the sequence is trivial: n Kn = {e}.

An approximated group is a pair (Γ, (Kn )) consisting of a discrete group together with a fixed approximating sequence. Here then is the construction of Higson, Lafforgue, and Skandalis that we will use. Definition 2.2. Let (Γ, (Kn )) be an approximated group, and for each n, let Γn := Γ/Kn be the associated quotient group, and πn : Γ → Γn the quotient map. For convenience, we also write Γ = Γ∞ and π∞ : Γ → Γ for the identity map. As a set, define G {n} × Γn . G := n∈N∪{∞}

Put the topology on G that is generated by the following open sets. (i) For each n ∈ N and g ∈ Γn , the singleton {(n, g)} is open. (ii) For each fixed g ∈ Γ and N ∈ N, the set {(n, πn (g)) | n ∈ N ∪ {∞}, n > N } is open. Finally equip G with the groupoid operations coming from the group structure on the ‘fibres’ over each n: precisely, the unit space is identified with the subspace {(n, g) ∈ G | g = e}; the source and range maps are defined by r(n, g) = s(n, g) = (n, e); and composition and inverse are defined by (n, g)(n, h) := (h, gh) and (n, g)−1 := (n, g −1 ). It is not difficult to check that G as constructed above is a locally compact, Hausdorff, second countable, ´etale groupoid. Moreover the unit space G(0) of G naturally identifies with the one-point compactification of N. We call G the HLS groupoid associated to the approximated group (Γ, (Kn )). In the rest of this section, we will characterise precisely when HLS groupoids as above are amenable, and when their maximal and reduced groupoid C ∗ -algebras coincide. We will then use results of Lubotzky and Shalom to show that examples that are ‘between’ these two properties exist, thus completing the proof of Theorem 1.2. Out first lemma characterises when an HLS groupoid G is amenable. We recall definitions of amenability that are appropriate for our purposes: for the groupoid definition, compare [3, Definition 5.6.13] and [1, Proposition 2.2.13 (ii)]; for the group definition, see for example [3, Definition 2.6.3 and Theorem 2.6.8].

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Definition 2.3. Let G be a locally compact, Hausdorff, ´etale groupoid. G is amenable if for any compact subset K of G and ǫ > 0 there exists a continuous, compactly supported function η : G → [0, 1] such that for all g ∈ K X X |η(h) − η(hg)| < ǫ. η(h) − 1 < ǫ and h∈G:s(h)=r(g)

h∈G:s(h)=r(g)

Let Γ be a discrete group. Γ is amenable if for any finite subset F of G and δ > 0 there exists a finitely supported function ξ : Γ → [0, 1] such that for all g ∈ F X |ξ(hg) − ξ(h)| < δ. h∈Γ

Lemma 2.4. Let G be the HLS groupoid associated to an approximated group (Γ, (Kn )). Then G is (topologically) amenable if and only if Γ is amenable. Proof. This is immediate from [1, Examples 5.1.3 (1)] or [14, Theorem 3.5]. For the convenience of the reader, however, we also provide a direct proof. Assume G is amenable, and let a finite subset F of Γ and δ > 0 be given. Let K be the finite (hence compact) subset {∞} × F of G, and let η : G → [0, 1] be as in the definition of amenability P for G for the compact subset K and error tolerance ǫ < δ/(1 + δ). Write M = g∈Γ η(∞, g) (and note that this is at least 1 − ǫ) and define ξ : Γ → [0, 1] by 1 ξ(g) = η(∞, g); M it is not difficult to check this works. Conversely, assume that Γ is amenable, and let a compact subset K of G and ǫ > 0 be given. Let F be a finite subset of Γ such that {n} × πn (F ) ⊇ K ∩ {n} × Γn for all n (compactness of K implies that such a set exists), and let ξ : Γ → [0, 1] be as in the definition of amenability for this F and error tolerance δ = ǫ. Define η : G → [0, 1] by X η(n, g) = ξ(h); −1 h∈πn (g)

it is again not difficult to check that this works.



Our next goal is to characterise when the maximal and reduced groupoid C ∗ algebras of an HLS groupoid G are equal. General definitions of the maximal and reduced C ∗ -algebras of an ´etale groupoid can be found in [13, Definition 2.3.18] or [3, pages 159 – 160]; the next definition specialises these to HLS groupoids. Definition 2.5. Let G be an HLS groupoid associated to the approximated group (Γ, (Kn )). Let Cc (G) denote the space of continuous, compactly supported, complexvalued functions on G equipped with the convolution product and involution defined by X f1 (n, gh−1 )f2 (n, h), f ∗ (n, g) := f (n, g −1 ). (f1 f2 )(n, g) := h∈Γn

∗

∗ The maximal C -algebra of G, denoted Cmax (G), is the completion of Cc (G) for the norm

kf kmax := sup{kρ(f )k | ρ : Cc (G) → B(H) a ∗-homomorphism }.

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For n ∈ N ∪ {∞}, define a ∗-representation ρn : Cc (G) → B(l2 (Γn )) by the formula X f (n, gh−1 )ξ(h). (ρn (f )ξ)(g) = h∈Γn

∗

∗ The reduced C -algebra of G, denoted Cred (G), is the completion of Cc (G) for the norm

kf kred := sup{kρn (f )k | n ∈ N ∪ {∞}}. Consider now the quotient ∗-homomorphism (1)

ψ : Cc (G) → C[Γ]

defined by restriction to the fibre at infinity. Let GN denote the open subgroupoid of G consisting of all pairs (n, g) with n ∈ N. The kernel of ψ is equal to the ∗-algebra Cc,0 (GN ) := {f ∈ Cc (G) | f (∞, g) = 0 for all g ∈ Γ}. ∗ ∗ Before proving our characterisation of when Cmax (G) = Cred (G), we will need the following ancillary lemma.

Lemma 2.6. The ∗-algebra Cc,0 (GN ) has a unique C ∗L -algebra completion, which ∗ canonically identifies with the C ∗ -algebraic direct sum n Cred (Γn ) (equivalently, ∗ ∗ with the reduced groupoid C -algebra Cred (GN )). Proof. Note first that L the ∗-subalgebra Cc (GN ) of Cc,0 (GN ) is isomorphic to the algebraic direct sum n∈N C[Γn ] of group algebras; in particular, it is isomorphic to an algebraic direct sum of matrix so has a unique C ∗ -algebra L completion: L algebras, ∗ ∗ ∗ (Γn ) in the C -algebraic direct sum n Cred (Γn ). For brevity, write A for n Cred the rest of the proof. The ∗-algebra Cc,0 (GN ) contains the commutative C ∗ -algebra C0 (N) as a ∗subalgebra, and therefore any C ∗ -norm on Cc,0 (GN ) restricts to the usual supremum norm on C0 (N). Fix g ∈ Γ, and say that f ∈ Cc,0 (GN ) is supported in the subset {(n, πn (g)) ∈ G | n ∈ N}. Then for any C ∗ -norm on Cc,0 (G), (2)

kf ∗ f k = kn 7→ |f (n, πn (g))|2 kC0 (N) = sup |f (n, πn (g))|2 . n∈N

Let now f be an arbitrary element of Cc,0 (GN ). As f is in Cc (G), there is a finite subset F of Γ such that f is supported in {(n, g) ∈ G | g ∈ πn (F )}. For fixed N ∈ N, write fN for the restriction of f to {(n, g) ∈ G | n ≤ N }. For any C ∗ -algebra norm k · k on Cc,0 (GN ), it follows from line (2) that kf − fN k ≤ |F | sup sup |f (n, g)| n>N g∈Γn

and therefore (as f is in Cc,0 (GN )), we have that kf − fN k → 0 as N → ∞. On the other hand, fN is in Cc (GN ) and therefore kfN k = kfN kA by uniqueness of the C ∗ -algebra norm on Cc (GN ). In particular (fN )∞ N =1 is Cauchy in the A norm, and its limit in A identifies naturally with f . It follows that kf kA makes sense, and that kf k = kf kA, completing the proof. 

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Lemma 2.7. Let G be the HLS groupoid associated to an approximated group (Γ, (Kn )). For each n ∈ N ∪ {∞}, let λn : C[Γ] → B(l2 (Γn )) denote the quasi-regular representation induced by the left multiplication action of Γ on Γn . ∗ ∗ Then Cmax (G) = Cred (G) if and only if the maximal norm on C[Γ] equals the norm defined by (3)

kxk :=

sup

kλn (x)k

n∈N∪{∞}

Proof. Let ψ : Cc (G) → C[Γ] be the quotient ∗-homomorphism from line (1). The ∗ ∗ (G) norms on Cc (G) induce the C ∗ -algebra norms Cmax (G) and Cred kxk∞,max := inf{kf kmax | f ∈ ψ −1 (x)} kxk∞,red := inf{kf kred | f ∈ ψ −1 (x)} ∗ ∗ on C[Γ]. These norms give rise to completions C∞,max (Γ) and C∞,red (Γ) of C[Γ] respectively; moreover, as ∗-representations of C[Γ] pull back to ∗-representations of ∗ ∗ Cc (G) via the map ψ : Cc (G) → C[Γ], it is immediate that C∞,max (Γ) = Cmax (Γ). Putting this together with Lemma 2.6, there is a commutative diagram of C ∗ algebras ∗ / C ∗ (Γ) /0, / C ∗ (G) /L (4) 0 max max n∈N Cred (Γn )

0

/

L

n∈N

∗ Cred (Γn )

 / C ∗ (G) red

 / C∗ ∞,red (Γ)

/0

where the vertical maps are the canonical quotients induced by the identity map on Cc (G) and C[Γ], and where the horizontal lines are both exact3. The five lemma ∗ ∗ ∗ ∗ then gives that Cmax (G) = Cred (G) if and only if Cmax (Γ) = C∞,red (Γ); to complete the proof, we must therefore show that for any x ∈ C[Γ] we have kxk∞,red =

sup

kλn (x)k.

n∈N∪{∞}

Fix an element x ∈ C[Γ]. For some suitably large N and all n ≥ N , the quotient map πn : Γ → Γn is injective on the support of x. Define f to be the element of Cc (G) given by  x(h) n ≥ N or n = ∞, and g = πn (h) for h ∈ supp(x) f (n, g) = , 0 n 0 there is a uniform bound on the cardinality of all r-balls, and is uniformly discrete if there is an absolute lower bound on the distance between any two points. Associated to such ∗ ∗ an X are maximal and reduced uniform Roe algebras Cu,max (X) and Cu,red (X). In coarse geometry, the natural analogue of the question addressed in this note is: ∗ ∗ is it possible that Cu,max (X) = Cu,red (X) for some space X without Yu’s property A [19, Section 2]? Indeed, Skandalis, Tu, and Yu [17] have associated to such X a coarse groupoid G(X) which is topologically amenable if and only if X has property A [17, Theorem 5.3]. Moreover, the maximal and reduced uniform Roe algebras of X identify naturally with the maximal and reduced groupoid C ∗ -algebras of G(X). This note grew out of an attempt to understand this question for the specific example looked at in [18, Example 1.15]. We were not, however, able to make any progress on the coarse geometric special case of the general groupoid question. Remark 3.6. A groupoid is principal if the only elements with the same source and range are the units. It is natural to ask if examples with the sort of properties in Theorem 1.2 are possible for principal groupoids: the coarse groupoids mentioned above are a special case, as are many other interesting examples coming from equivalence relations and free actions of groups. Again, our results seem to shed no light on this question, and we do not know enough to speculate either way. References [1] C. Anantharaman-Delaroche and J. Renault. Amenable groupoids. L’enseignment Math´ ematique, 2000. [2] B. Bekka. On the full C ∗ -algebras of arithmetic groups and the congruence subgroup problem. Forum Math., 11(6):705–715, 1999. [3] N. Brown and N. Ozawa. C ∗ -Algebras and Finite-Dimensional Approximations, volume 88 of Graduate Studies in Mathematics. American Mathematical Society, 2008. [4] N. Higson, V. Lafforgue, and G. Skandalis. Counterexamples to the Baum-Connes conjecture. Geom. Funct. Anal., 12:330–354, 2002. [5] A. Hulanicki. Means and Følner conditions on locally compact groups. Studia Mathematica, 27:87–104, 1966. [6] D. Kerr. C ∗ -algebras and topological dynamics: finite approximations and paradoxicality. Available on the author’s website, 2011. [7] A. Lubotzky. Discrete Groups, Expanding Graphs and Invariant Measures. Birkh¨ auser, 1994. [8] A. Lubotzky and Y. Shalom. Finite representations in the unitary dual and Ramanujan groups. In Discrete geometric analysis, number 347 in Contemporary Mathematics, pages 173–189. American Mathematical Society, 2004. [9] M. Matsumura. A characterization of amenability of group actions on C ∗ -algebras. J. Operator Theory, 72(1):41–47, 2014. [10] P. Muhly, J. Renault, and D. Williams. Continuous-trace groupoid C ∗ -algebras. III. Trans. Amer. Math. Soc., 348(9):3621–3641, 1996. [11] J. Renault. A groupoid approach to C ∗ -algebras, volume 793 of Lecture Notes in Mathematics. Springer-Verlag, 1980. [12] J. Renault. The ideal structure of groupoid crossed product C ∗ -algebras. J. Operator Theory, 25:3–36, 1991. [13] J. Renault. C ∗ -algebras and dynamical systems. Publica¸co ˜es Mathem´ aticas do IMPA, 27◦ Col´ oquio Brasilieiro de Mathem´ atica. Instituto Nacional de Matem´ atica Pura e Aplicada, 2009.

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[14] J. Renault. Topological amenability is a Borel property. arXiv:1302.0636, to appear in Math. Scand., 2013. [15] A. Selberg. On the estimation of Fourier coefficients of modular forms. In Proc. Sympos. Pure Math., volume VIII, pages 1–15, 1965. [16] A. Sims and D. Williams. Amenability for Fell bundles over groupoids. Illinois J. Math, 67:429–444, 2013. [17] G. Skandalis, J.-L. Tu, and G. Yu. The coarse Baum-Connes conjecture and groupoids. Topology, 41:807–834, 2002. ˇ [18] J. Spakula and R. Willett. Maximal and reduced Roe algebras of coarsely embeddable spaces. J. Reine Angew. Math., 678:35–68, 2013. [19] G. Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math., 139(1):201–240, 2000. ¯ noa, Department of Mathematics, 2565 Rufus Willett, University of Hawai‘i at Ma McCarthy Mall, Honolulu, HI 96822-2273, USA E-mail address: [email protected] URL: http://math.hawaii.edu/~rufus/